Actions of automorphism groups of free groups on spaces of Jacobi diagrams. I
AACTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPSON SPACES OF JACOBI DIAGRAMS. I
MAI KATADA
Abstract.
We study a filtered vector space A d ( n ) over a field k of charac-teristic 0, which consists of Jacobi diagrams of degree d on n oriented arcs foreach n, d ≥
0. We consider an action of the automorphism group Aut( F n )of the free group F n of rank n on the space A d ( n ), which is induced by anaction of handlebody groups on bottom tangles. The action of Aut( F n ) on A d ( n ) induces an action of the general linear group GL( n, k ) on the associatedgraded vector space of A d ( n ), which is regarded as the vector space B d ( n )consisting of open Jacobi diagrams. Moreover, the Aut( F n )-action on A d ( n )induces an action on B d ( n ) of the associated graded Lie algebra gr(IA( n ))of the IA-automorphism group IA( n ) of F n with respect to its lower centralseries. We use an irreducible decomposition of B d ( n ) and computation of thegr(IA( n ))-action on B d ( n ) to study the Aut( F n )-module structure of A d ( n ).In particular, we consider the case where d = 2 in detail and give an inde-composable decomposition of A ( n ). We also consider a functor A d , whichincludes the Aut( F n )-module structure of A d ( n ) for all n . Contents
1. Introduction 12. Preliminaries 63. Functors A d and B d A and B n )) on B d ( n ) 166. Aut( F n )-module structure of A ( n ) and indecomposable decompositionof the functor A Introduction
The
Kontsevich integral is a universal finite type invariant for links [15, 2]. Sinceany links are obtained from string links or bottom tangles via closure, it is naturalto consider the Kontsevich integral for string links [7, 3] and bottom tangles [8].The target space of the Kontsevich integral for string links and bottom tanglesconsists of Jacobi diagrams on oriented arcs.
Date : February 5, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Jacobi diagrams, Automorphism groups of free groups, General lineargroups, IA-automorphism groups of free groups. a r X i v : . [ m a t h . QA ] F e b MAI KATADA
Habiro and Massuyeau [10] extended the Kontsevich integral to construct a func-tor Z : B → ˆ A from the category B of bottom tangles in handlebodies to the degreecompletion ˆ A of the category A of Jacobi diagrams in handlebodies. For each object n ∈ N of B , the automorphism group Aut B ( n ) of n is isomorphic to the handlebodygroup H n = Homeo( U n , D ) / (isotopy rel D ) , where U n is a genus n handlebody obtained from a cube by attaching n handleson the top square, and D is the bottom square. Thus, H n acts on the set B (0 , n )of n -component bottom tangles in the cube U . An analogous action of H n onˆ A (0 , n ) can be obtained by the functor Z . Fundamental group gives a surjectivegroup homomorphism H n (cid:16) Aut( F n ) , where Aut( F n ) is the automorphism groupof the free group F n = (cid:104) x , · · · , x n (cid:105) of rank n . Then, the action of H n on ˆ A (0 , n )induces an action of Aut( F n ) on the k -vector space A d ( n ) of Jacobi diagrams ofdegree d on n oriented arcs for any n, d ≥
0, where k is a field of characteristic 0.The aim of the present paper is to study the Aut( F n )-module structure of thespace A d ( n ). We consider A d ( n ) as a filtered vector space, whose associated gradedvector space is isomorphic to the graded vector space B d ( n ) of colored open Jacobidiagrams . The action of Aut( F n ) on A d ( n ) induces an action on B d ( n ) of the generallinear group GL( n ; k ) of degree n and an action on B d ( n ) of the associated gradedLie algebra gr(IA( n )) of the IA-automorphism group
IA( n ) of F n with respect toits lower central series. To study the Aut( F n )-module structure of A d ( n ), we usean irreducible decomposition of the GL( n ; k )-module B d ( n ) and the gr(IA( n ))-action on B d ( n ). In particular, we consider the Aut( F n )-module structure of A ( n )and finally we give an indecomposable decomposition of A ( n ). We also considera functor A d from the opposite category of the category of finitely generated freegroups to the category of filtered vector spaces, which includes the Aut( F n )-modulestructure of A d ( n ) for any n ≥ The space A d ( n ) of Jacobi diagrams. We work over a fixed field k ofcharacteristic 0. We consider here the k -vector space A d ( n ) of Jacobi diagrams onoriented arcs, which is the main object of the present paper.For n ≥
0, let X n = 1 n · · · n arc components. The k -vector space A d ( n ) is spanned by Jacobi diagrams on X n of degree d modulo the STU relations. Here the degree of Jacobi diagramsis defined to be half the number of vertices as usual. (See Section 2.1 for furtherdetails.)We consider a filtration for A d ( n ) A d ( n ) = A d, ( n ) ⊃ A d, ( n ) ⊃ · · · ⊃ A d, d − ( n ) ⊃ A d, d − ( n ) = 0 , such that A d,k ( n ) ⊂ A d ( n ) is the subspace spanned by Jacobi diagrams with atleast k trivalent vertices. Hence, A d ( n ) is a filtered vector space. For example,1 2 n · · · ∈ A , ( n ) , n · · · ∈ A , ( n ) , n · · · ∈ A , ( n ) . A functor A d and an Aut( F n ) -action on A d ( n ) . We construct a functor A d : F op → fVect CTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON JACOBI DIAGRAMS 3 from the opposite category F op of the category F of finitely generated free groupsto the category fVect of filtered vector spaces over k , which maps F n to the filteredvector space A d ( n ). The functor A d gives a mapHom( F m , F n ) × A d ( n ) → A d ( m )for m, n ≥
0. (See Section 3.2 for the definition.) For example, for an element f ∈ Hom( F , F ) defined by f ( x ) = x x , f ( x ) = x x , we have f · . When n = m , we have an action of the opposite of the endomorphism monoidEnd( F n ) op on A d ( n ) End( F n ) op × A d ( n ) → A d ( n ) . By restricting this action to the automorphism group Aut( F n ) ∼ = Aut( F n ) op , weobtain an action of Aut( F n ) on A d ( n ). Here, we identify Aut( F n ) with its oppositegroup Aut( F n ) op by taking the inverse of each element. The Aut( F n )-action on A d ( n ) induces an action on A d ( n ) of the outer automorphism group Out( F n ) of F n .(See Theorem 5.1.)1.3. A functor B d and a GL( n ; Z ) -action on B d ( n ) . The associated gradedvector space gr( A d ( n )) of the filtered vector space A d ( n ) can be identified with thevector space B d ( n ) of colored open Jacobi diagrams, which we explain below, viathe PBW map [2, 3]. The Aut( F n )-action on A d ( n ) induces an action of GL( n ; Z )on B d ( n ). In order to consider the Aut( F n )-module structure of A d ( n ), we use anirreducible decomposition of the GL( n ; Z )-module B d ( n ), which can be obtainedby using results by Bar-Natan [4] for small d .For n ≥
0, let V n = (cid:76) ni =1 k v i be an n -dimensional k -vector space. The k -vectorspace B d ( n ) is spanned by V n -colored open Jacobi diagrams of degree d modulothe AS, IHX and multilinearity relations, where “ V n -colored” means that eachunivalent vertex is colored by an element of V n . (See Section 3.3 for further details.)We consider a grading for B d ( n ) such that the degree k part B d,k ( n ) ⊂ B d ( n ) isspanned by open Jacobi diagrams with exactly k trivalent vertices. For example, v v v v ∈ B , ( n ) , v v v ∈ B , ( n ) , v v ∈ B , ( n ) . The functor B d : FAb op → gVect from the opposite category FAb op of the category FAb of finitely generated freeabelian groups to the category gVect of graded vector spaces over k maps eachobject Z n of FAb op to the graded vector space B d ( n ). Here, we have a mapHom( Z m , Z n ) × B d ( n ) → B d ( m )for m, n ≥
0, which is given by matrix multiplication on each coloring. (See Section3.3 for the definition.)When n = m , we have an action of the opposite of the endomorphism monoidEnd( Z n ) op on B d ( n ) End( Z n ) op × B d ( n ) → B d ( n ) . MAI KATADA
By restricting this action to GL( n ; Z ) ∼ = GL( n ; Z ) op , we obtain an action of GL( n ; Z )on B d ( n ). Moreover, the GL( n ; Z )-action on B d ( n ) extends to an action of GL( n ; k )on B d ( n ).Recall that for any partition λ of N ≥ n rows, the Schur functor S λ gives a simple GL( V n )-module S λ V n = V ⊗ Nn · c λ , where c λ ∈ k S N is the Youngsymmetrizer corresponding to λ . We have irreducible decompositions of GL( V n )-modules B ( n ) = B , ( n ) ∼ = S (2) V n , (1) B ( n ) = B , ( n ) ⊕ B , ( n ) ⊕ B , ( n ) ∼ = ( S (4) V n ⊕ S (2 , V n ) ⊕ S (1 , , V n ⊕ S (2) V n . (2)We observe that the functor A d induces the functor B d . Let ab : F → FAb denote the abelianization functor, and ab op : F op → FAb op its opposite functor.Let gr : fVect → gVect denote the functor that sends a filtered vector space to itsassociated graded vector space. Proposition 1.1 (see Proposition 3.2) . For d ≥ , there is a natural isomorphism θ d : gr ◦ A d ∼ = ⇒ B d ◦ ab op . In diagram, we have F op A d (cid:47) (cid:47) ab op (cid:15) (cid:15) fVect gr (cid:15) (cid:15) FAb op B d (cid:47) (cid:47) gVect . ∼ = (cid:119)(cid:127) θ d The functor A . Here we consider the functors A and B . By Proposition1.1, we have A ∼ = B ◦ ab op . Since we have isomorphisms of Aut( F n )-modules A ( n ) ∼ = B ( n ) ∼ = S (2) V n = Sym ( V n )by (1), A ( n ) is simple for any n ≥
1. It follows that the functor A is indecom-posable.1.5. An action of gr(IA( n )) on the space B d ( n ) . Let IA( n ) denote the IA-automorphism group of F n , which is the kernel of the canonical homomorphismAut( F n ) → Aut( H ( F n ; Z )) ∼ = GL( n ; Z ). Let Γ ∗ (IA( n )) = (Γ r (IA( n ))) r ≥ denotethe lower central series of IA( n ), and gr(IA( n )) = (cid:76) r ≥ gr r (IA( n )) the associatedgraded Lie algebra, where gr r (IA( n )) = Γ r (IA( n )) / Γ r +1 (IA( n )).To study the Aut( F n )-module structure of A d ( n ), we use an action of gr(IA( n ))on the graded vector space B d ( n ) ∼ = gr( A d ( n )). Theorem 1.2 (see Proposition 5.9 and Corollary 5.15) . There is an action of thegraded Lie algebra gr(IA( n )) on the graded vector space B d ( n ) , which consists of GL( n ; Z ) -module homomorphisms B d,k ( n ) ⊗ Z gr r (IA( n )) → B d,k + r ( n ) for k ≥ and r ≥ . CTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON JACOBI DIAGRAMS 5
The functor A . Here we give a direct decomposition of the functor A .We use the graphical notation = 1 2 ∈ A (2). Set P (cid:48) = sym , P (cid:48)(cid:48) = alt alt ∈ A (4) , where sym corresponds to the Young symmetrizer c (4) and alt corresponds to theYoung symmetrizer c (1 , . (See Section 6.2 for further details.) Let A (cid:48) (resp. A (cid:48)(cid:48) ) : F op → fVect be the subfunctor of the functor A such that A (cid:48) ( n ) := Span k { A ( f )( P (cid:48) ) : f ∈ F op (4 , n ) } ⊂ A ( n )(resp. A (cid:48)(cid:48) ( n ) := Span k { A ( f )( P (cid:48)(cid:48) ) : f ∈ F op (4 , n ) } ⊂ A ( n )) . Proposition 1.3 (see Proposition 6.5) . We have a direct decomposition A = A (cid:48) ⊕ A (cid:48)(cid:48) in the functor category fVect F op ; that is, we have a decomposition of filtered vectorspaces (3) A ( n ) = A (cid:48) ( n ) ⊕ A (cid:48)(cid:48) ( n ) for any n ≥ and a decomposition of filter-preserving maps A ( f ) = A (cid:48) ( f ) ⊕ A (cid:48)(cid:48) ( f ) for any morphism f in F op . F n ) -module structure of A ( n ) and indecomposable decomposi-tion of A . Here, we consider the Aut( F n )-module structure of A ( n ) and give anindecomposable decomposition of the functor A .The direct decomposition (3) of A ( n ) induces a direct decomposition of B ( n ) B ( n ) = B (cid:48) ( n ) ⊕ B (cid:48)(cid:48) ( n ) . We have B (cid:48)(cid:48) ( n ) = B (cid:48)(cid:48) , ( n ) ⊕ B , ( n ) ⊕ B , ( n ) , where B (cid:48)(cid:48) , ( n ) = B (cid:48)(cid:48) ( n ) ∩ B , ( n ).By (2), we have the following irreducible decompositions of GL( V n )-modules B (cid:48) ( n ) ∼ = S (4) V n ,B (cid:48)(cid:48) ( n ) = B (cid:48)(cid:48) , ( n ) ⊕ B , ( n ) ⊕ B , ( n ) ∼ = S (2 , V n ⊕ S (1 , , V n ⊕ S (2) V n . In addition to the GL( V n )-module structure of B (cid:48)(cid:48) ( n ), we use the gr(IA( n ))-actionon B (cid:48)(cid:48) ( n ) to consider the Aut( F n )-module structure of A (cid:48)(cid:48) ( n ). Theorem 1.4 (see Theorem 6.9) . For n ≥ , we have an indecomposable decom-position of Aut( F n ) -modules A ( n ) = A (cid:48) ( n ) ⊕ A (cid:48)(cid:48) ( n ) . Here, A (cid:48) ( n ) is simple, and A (cid:48)(cid:48) ( n ) admits a unique composition series of length A (cid:48)(cid:48) ( n ) (cid:41) A , ( n ) (cid:41) A , ( n ) (cid:41) that is, A (cid:48)(cid:48) ( n ) has no nonzero proper Aut( F n ) -submodules other than A , ( n ) and A , ( n ) .(For n = 1 , , see Theorem 6.9.) MAI KATADA
Corollary 1.5 (see Corollary 6.11) . We have
Ext k Aut( F n ) ( S (2 , V n , S (1 , , V n ) (cid:54) = 0 , Ext k Aut( F n ) ( S (1 , , V n , S (2) V n ) (cid:54) = 0 for n ≥ . By using Theorem 1.4, we obtain an indecomposable decomposition of the func-tor A . Theorem 1.6 (see Theorem 6.14) . The direct decomposition A = A (cid:48) ⊕ A (cid:48)(cid:48) isindecomposable in the functor category fVect F op . In a paper in preparation [12], we plan to study the Aut( F n )-module structureof A d ( n ) and the functors A d for d ≥
3. See Section 7.1.8.
Organization of the paper.
In Section 2, we recall some notions and def-initions about Jacobi diagrams, open Jacobi diagrams and the category of Jacobidiagrams in handlebodies. In Section 3, we construct functors A d : F op → fVect and B d : FAb op → gVect and observe that A d induces B d . In Section 4, we com-pute the functors A and B explicitly. In Section 5, we define an action of thegraded Lie algebra gr(IA( n )) on the graded vector space B d ( n ). Here, we defineactions of extended graded Lie algebras on graded vector spaces. In Section 6,we consider the Aut( F n )-module structure of A ( n ) and give an indecomposabledecomposition of A . Finally, in Section 7, we explain the contexts of the paper inpreparation [12], in which we study the higher degree cases of d ≥ Acknowledgments.
The author would like to thank Kazuo Habiro for carefulreading and valuable advice. She would also like to thank Gw´ena¨el Massuyeau andSakie Suzuki for helpful comments.2.
Preliminaries
In this section, we recall some notions of Jacobi diagrams and open Jacobi di-agrams and the category A of Jacobi diagrams in handlebodies. In what follows,we work over a fixed field k of characteristic 0.2.1. Jacobi diagrams and open Jacobi diagrams.
In this section, we recallJacobi diagrams and open Jacobi diagrams defined in [2], [3] and [20].A uni-trivalent graph is a finite graph whose vertices are either univalent ortrivalent. A trivalent vertex is oriented if it has a fixed cyclic order of the threeedges around it. A vertex-oriented uni-trivalent graph is a uni-trivalent graph suchthat each trivalent vertex is oriented.For n ≥
0, let X n be the oriented 1-manifold consisting of n arc components asdepicted in Figure 1. 1 2 nX n = Figure 1.
The oriented 1-manifold X n . CTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON JACOBI DIAGRAMS 7 A Jacobi diagram on X n is a vertex-oriented uni-trivalent graph such that uni-valent vertices are embedded into the interior of X n and each connected componenthas at least one univalent vertex. Two Jacobi diagrams D and D (cid:48) on X n are re-garded as the same if there is a homeomorphism f : D ∪ X n → D (cid:48) ∪ X n whoserestriction to X n is isotopic to the identity map of X n . In figures, we depict X n assolid lines and Jacobi diagrams as dashed lines in such a way that each trivalentvertex is oriented in the counterclockwise order.Let A ( X n ) denote the k -vector space spanned by Jacobi diagrams on X n modulothe STU relation, which is described in Figure 2.= − Figure 2.
The STU relation.The degree of a Jacobi diagram is defined to be half the number of its vertices.Since the STU relation is homogeneous with respect to the degree, we have a grading A ( X n ) = (cid:77) d ≥ A d ( X n ) , where A d ( X n ) ⊂ A ( X n ) is the subspace spanned by Jacobi diagrams of degree d .For k ≥
0, let A d,k ( X n ) ⊂ A d ( X n ) be the subspace spanned by Jacobi diagramswith at least k trivalent vertices. We have A ( X n ) = A , ( X n ) ∼ = k for d = 0. For d ≥
1, we have a filtration A d ( X n ) = A d, ( X n ) ⊃ A d, ( X n ) ⊃ A d, ( X n ) ⊃ · · · ⊃ A d, d − ( X n ) = 0 . Note that we have A d, d − ( X n ) = 0 since a Jacobi diagram on X n with only oneunivalent vertex vanishes by using the STU relations. We consider the gradedvector space gr( A d ( X n )) := (cid:76) k ≥ gr k ( A d ( X n )) associated to the above filtration A d, ∗ ( X n ), where gr k ( A d ( X n )) := A d,k ( X n ) / A d,k +1 ( X n ).An open Jacobi diagram is a vertex-oriented uni-trivalent graph such that eachconnected component has at least one univalent vertex.Let T be a set. A T -colored open Jacobi diagram is an open Jacobi diagramsuch that each univalent vertex is colored by an element of T . In figures, we depict T -colored open Jacobi diagrams as solid lines in such a way that each trivalentvertex is oriented in the counterclockwise order.Let B ( T ) denote the k -vector space spanned by T -colored open Jacobi diagramsmodulo the AS and IHX relations, which are depicted in Figure 3.= − = − , Figure 3.
The AS and IHX relations.The degree of a T -colored open Jacobi diagram is defined to be half the numberof vertices. Since the AS and IHX relations are homogeneous with respect to the MAI KATADA degree, we have a grading B ( T ) = (cid:77) d ≥ B d ( T ) , where B d ( T ) ⊂ B ( T ) is the subspace spanned by T -colored open Jacobi diagramsof degree d .For k ≥
0, let B d,k ( T ) ⊂ B d ( T ) be the subspace spanned by open Jacobidiagrams with exactly k trivalent vertices. We have B ( T ) = B , ( T ) = k ∅ for d = 0. For d ≥
1, we have B d ( T ) = d − (cid:77) k =0 B d,k ( T ) . Note that B d,k ( T ) = 0 for k ≥ d since an open Jacobi diagram has at least oneunivalent vertex and for k = 2 d − n ] := { , · · · , n } ⊂ N . Bar-Natan [2, 3] proved that A ( X n ) is isomorphic to B ([ n ]). This is a diagrammaticinterpretation of the Poincare–Birkhoff–Witt theorem. Proposition 2.1 (PBW theorem [2, 3])) . For d ≥ , we have an isomorphism ofvector spaces χ d : B d ([ n ]) ∼ = −→ A d ( X n ) . If D ∈ B ([ n ]) is an [ n ] -colored open Jacobi diagram of degree d such that for any i ∈ [ n ] , D has k i univalent vertices colored by i , then χ ( D ) ∈ A ( X n ) is the averageof the (cid:81) i ∈ [ n ] ( k i )! ways of attaching the univalent vertices colored by i to the i -thcomponent of X n .Moreover, the map χ d induces an isomorphism χ d,k : B d,k ([ n ]) ∼ = −→ gr k ( A d, ∗ ( X n )) . Note that two Jacobi diagrams of A d,k ( X n ) appearing in the average of the (cid:81) i ∈ [ n ] ( k i )! ways are equivalent in the quotient space gr k ( A d, ∗ ( X n )) by the STUrelations. Therefore, the average of the (cid:81) i ∈ [ n ] ( k i )! ways of attaching univalentvertices coincides with an arbitrary way of attaching them in gr k ( A d, ∗ ( X n )).2.2. The category A of Jacobi diagrams in handlebodies.
Here we brieflyreview the category A of Jacobi diagrams in handlebodies defined in [10].The objects in A are nonnegative integers. To define the hom-set A ( m, n ), weneed the notion of ( m, n )-Jacobi diagrams, which we explain below.Let I = [ − , m ≥
0, let U m ⊂ R denote the handlebody of genus m thatis obtained from the cube I by attaching m handles on the top square I × { } asdepicted in Figure 4. We call l := I × { } × {− } the bottom line of U m . We call S := I × {− } the bottom square of U m . For i = 1 , · · · , m , let x i be a loop whichgoes through only the i -th handle of the handlebody U m just once and let x i denoteits homotopy class as well. In what follows, for loops γ and γ with base pointson l , let γ γ denote the loop that goes through γ first and then goes through γ .That is, we write a product of elements of the fundamental group of U m in theopposite order to the usual one. Let ¯ x i ∈ H ( U m ; k ) denote the homology class of x i . We have π ( U m ) = (cid:104) x , · · · , x m (cid:105) and H ( U m ; k ) = (cid:76) mi =1 k ¯ x i . CTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON JACOBI DIAGRAMS 9 S m · · · lx x m · · · Figure 4.
The handlebody U m .For m, n ≥
0, an ( m, n ) -Jacobi diagram consists of a Jacobi diagram D on X n and a map X n ∪ D → U m which maps ∂X n into the bottom line l of U m in such away that the endpoints of X n are uniformly distributed and that for i = 1 , · · · , n ,the i -th arc component of X n goes from the 2 i -th point to the 2 i − m, n )-Jacobi diagramsif they are homotopic in U m relative to the endpoints of X n . Figure 5 shows a(2 , D . For m, n ≥
0, the hom-set A ( m, n ) is the k -vector space D = : 2 → Figure 5.
A (2 , m, n )-Jacobi diagrams modulo the STU relations. We usually depict( m, n )-Jacobi diagrams by drawing their images under the orthogonal projection of R onto R × { } × R .We use the box notation to represent certain linear combinations of Jacobi dia-grams as depicted in Figure 6. Dashed lines and solid lines are allowed to go throughthe box and each of them appears as one summand. The sign of a summand cor-responding to a solid line is determined by the compatibility of its orientation withthe direction of the box. A summand corresponding to a dashed line has a positivesign and the orientation of the new trivalent vertex is determined by the directionof the box. See [10] for the definition.For D : m → n and D (cid:48) : p → m , the composition D ◦ D (cid:48) is defined as follows. Byusing isotopies of U m , we can transform D into an ( m, n )-Jacobi diagram ˜ D eachof whose handle has only solid and dashed lines parallel to the handle core. Thecomposition D ◦ D (cid:48) is obtained by stacking on the top of ˜ D a suitable cabling of D (cid:48) that matches the source of ˜ D . Figure 7 shows the composition D ◦ D (cid:48) of the = − + · · · ++ =: Figure 6.
The box notation.(2 , D , which is given in Figure 5, and a (3 , D (cid:48) . D (cid:48) = D ◦ D (cid:48) = Figure 7.
The composition D ◦ D (cid:48) .The degree of an ( m, n )-Jacobi diagram is the degree of its Jacobi diagram. Let A d ( m, n ) ⊂ A ( m, n ) be the subspace spanned by ( m, n )-Jacobi diagrams of degree d . We have A ( m, n ) = (cid:76) d ≥ A d ( m, n ).The category A has a structure of a linear symmetric strict monoidal category.See [11] for the definition of symmetric strict monoidal categories. The tensorproduct on objects is addition. The monoidal unit is 0. The tensor product onmorphisms is juxtaposition followed by horizontal rescaling and relabelling of in-dices. For example, Figure 8 shows the tensor product of a (1 , , P , : 2 → ⊗ Figure 8.
The tensor product.depicted in Figure 9.In this paper, we mainly consider A d (0 , n ) for d, n ≥
0. Note that A d (0 , n ) ∼ = A d ( X n ) as k -vector spaces. 3. Functors A d and B d In this section, we define a functor A d : F op → fVect from the opposite category F op of the category F of finitely generated free groups to the category fVect of CTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON JACOBI DIAGRAMS 11 P , = Figure 9.
The symmetry.filtered vector spaces over k . We define another functor B d : FAb op → gVect fromthe opposite category FAb op of the category FAb of finitely generated free abeliangroups to the category gVect of graded vector spaces over k . We prove that thefunctor A d induces the functor B d .3.1. The categories F, FAb, fVect and gVect.
Let us start with the definitionsof the categories F , FAb , fVect and gVect .For n ≥
0, let F n = (cid:104) x , · · · , x n (cid:105) be the free group of rank n . The category F offinitely generated free groups is the full subcategory of the category Grp of groupssuch that the class of objects is { F n : n ≥ } . We identify the object F n with theinteger n . Thus, F ( n, m ) = Hom( F n , F m ) ∼ = F nm . The category F is a symmetricstrict monoidal category.The category FAb of finitely generated free abelian groups is the full subcategoryof the category Ab of abelian groups such that the class of objects is { Z n : n ≥ } .We identify the object Z n with the integer n . Thus, FAb ( n, m ) = Hom( Z n , Z m ) ∼ =Mat( m, n ; Z ). The category FAb is also a symmetric strict monoidal category.Let ab : F → FAb denote the restriction of the abelianization functor ab :
Grp → Ab . Here the functor ab maps F n to Z n in a canonical way. In thefollowing sections, we use the opposite functor ab op : F op → FAb op .Let fVect denote the category of filtered vector spaces and filter-preservingmorphisms. A filtered vector space is a k -vector space V with a decreasing sequenceof vector spaces V = V ⊃ V ⊃ · · · .Let gVect denote the category of graded vector spaces and degree-preservingmorphisms. A graded vector space is a k -vector space W = (cid:76) d ≥ W d .For a filtered vector space V , set gr d ( V ) := V d /V d +1 for d ≥
0. We call gr( V ) := (cid:76) d ≥ gr d ( V ) the associated graded vector space of V . Let gr : fVect → gVect be the functor that sends a filtered vector space V to the associated graded vectorspace gr( V ) and a filter-preserving morphism f : V → W to a degree-preservingmorphism gr( f ) : gr( V ) → gr( W ) defined by gr( f )([ v ] V d +1 ) = [ f ( v )] W d +1 for v ∈ V d .3.2. The functor A d : F op → fVect. We define a functor A d : F op → fVect .Let d, n ≥
0. Set A d ( n ) := A d (0 , n ) ∼ = A d ( X n ) . For k ≥
0, let A d,k ( n ) ⊂ A d ( n ) be the subspace spanned by Jacobi diagrams withat least k trivalent vertices. We have an isomorphism A d,k ( n ) ∼ = A d,k ( X n ) . Thus, we have A ( n ) = A , ( n ) ∼ = k . For d ≥
1, we have a filtration A d ( n ) = A d, ( n ) ⊃ A d, ( n ) ⊃ A d, ( n ) ⊃ · · · ⊃ A d, d − ( n ) = 0 . Let k F be the linearization of the category F . Here, the class of objects in k F is the same as that in F and the hom-set k F ( m, n ) is the k -vector space spannedby all of the morphisms m → n in F for m, n ≥
0. We have an isomorphism k F op ( m, n ) ∼ = −→ A ( m, n ) of k -vector spaces [10]. Note that A ( m, n ) = k { ( m, n )-Jacobi diagrams with empty Jacobi diagram } = k { homotopy classes of maps X n → U m relative to the boundary } . For a map f : X n → U m such that f ( ∂X n ) ⊂ l , let ˜ f = f ∪ id l : X n ∪ l → U m and ˜ f ∗ : π ( X n ∪ l ) ∼ = F n → π ( U m ) ∼ = F m be the induced map on the fundamental groups.The linear map A ( m, n ) → k F op ( m, n ) that sends f to ˜ f ∗ is an isomorphism.We define a map A d : F op ( m, n ) → fVect ( A d ( m ) , A d ( n ))by A d : F op ( m, n ) (cid:44) → k F op ( m, n ) ∼ = −→ A ( m, n ) ◦ −→ fVect ( A d ( m ) , A d ( n )) . Note that since any element of A ( m, n ) has an empty Jacobi diagram, the compo-sition of an element of A ( m, n ) with an element of A d ( m ) preserves the filtration.It can be easily checked that A d is a functor.3.3. The functor B d : FAb op → gVect. In this section, we define a functor B d : FAb op → gVect .Let V n := H ( U n ; k ) = Hom( H ( U n ; k ) , k )and let { v i } denote the dual basis of { ¯ x i } . We have V n = (cid:76) ni =1 k v i .Let B d ( n ) denote the k -vector space spanned by V n -colored open Jacobi diagramsof degree d modulo the AS, IHX and multilinearity relations, where the multilin-earity relation is shown in Figure 10. Since V n = (cid:76) ni =1 k v i , the space B d ( n ) isisomorphic to the space B d ([ n ]) defined in Section 2.1. aw + bw = a w + b w for a, b ∈ k , w , w ∈ V n . Figure 10.
Multilinearity.For k ≥
0, let B d,k ( n ) ⊂ B d ( n ) be the subspace spanned by open Jacobi diagramswith exactly k trivalent vertices. We have an isomorphism B d,k ( n ) ∼ = B d,k ([ n ]) . Thus, we have B ( n ) = B , ( n ) = k ∅ . For d ≥
1, we have a grading B d ( n ) = d − (cid:77) k =0 B d,k ( n ) . CTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON JACOBI DIAGRAMS 13
Let T be a finite set. A T -colored open Jacobi diagram D is called special if themap { univalent vertices of D } → T that gives the coloring of D is a bijection.Define D d,k as the k -vector space spanned by special [2 d − k ]-colored open Jacobidiagrams of degree d modulo the AS and IHX relations. The space D d,k has an S d − k action given by the action on the colorings. Considering V ⊗ d − kn as a right S d − k -module by the action which permutes the factors, we have an isomorphism(4) B d,k ( n ) ∼ = V ⊗ d − kn ⊗ k S d − k D d,k . Thus, any element of B d,k ( n ) can be written in the form u ( w , · · · , w d − k ) := ( w ⊗ · · · ⊗ w d − k ) ⊗ u for u ∈ D d,k and w , · · · , w d − k ∈ V n .For m, n ≥
0, we define a map B d : FAb op ( m, n ) → gVect ( B d ( m ) , B d ( n ))as follows. We consider an element of FAb op ( m, n ) = Mat( m, n ; Z ) as an m × n matrix and an element of V n as a 1 × n matrix. For example, we consider v i ∈ V n as the i -th standard basis. For f ∈ FAb op ( m, n ) and u ( w , · · · , w d − k ) ∈ B d ( m ),we define B d ( f )( u ( w , · · · , w d − k )) := u ( w · f, · · · , w d − k · f ) . It can be easily seen that B d is a functor.We can apply the definition of B d : FAb op ( m, n ) → gVect ( B d ( m ) , B d ( n )) tothe opposite group GL( n ; k ) op of the general linear group GL( n ; k ) with coefficientin k to obtain a group homomorphism B d : GL( n ; k ) op → Aut gVect ( B d ( n )) . Then we have a GL( n ; k )-action on B d ( n ) by identifying GL( n ; k ) with GL( n ; k ) op by taking the inverse of each element.On the other hand, we consider the GL( V n )-action on B d ( n ) that is determinedby the standard action of GL( V n ) on each coloring. Here, we consider an elementof V n = (cid:76) ni =1 k v i as an n × n ; k )-action on B d ( n ) factors through the dual action of GL( n ; k ) on V n and the standard actionof GL( V n ) on B d ( n ):(5) GL( n ; k ) t ( · ) − −−−→ GL( V n ) → Aut gVect ( B d ( n )) . Note that the isomorphism (4) is a GL( V n )-module isomorphism.3.4. Relation between the functors A d and B d . In this section, we show thatthe functor A d defined in Section 3.2 induces the functor B d defined in Section 3.3.In the following lemma, we observe that we can identify the associated gradedvector space gr( A d ( n )) of the filtered vector space A d ( n ) with the graded vectorspace B d ( n ). Lemma 3.1.
For d, n, k ≥ , we have an isomorphism of k -vector spaces θ d,n,k : gr k ( A d ( n )) ∼ = −→ B d,k ( n ) , which maps a Jacobi diagram D on X n to an open Jacobi diagram θ d,n,k ( D ) thatis obtained from D by assigning the color v i to a univalent vertex which is attachedto the i -th arc component of X n for any i = 1 , · · · , n . Taking direct sum, we have an isomorphism of graded vector spaces θ d,n : gr( A d ( n )) ∼ = −→ B d ( n ) . We call θ d,n the PBW map.Proof. By identifying B d,k ( n ) with B d,k ([ n ]) and A d,k ( n ) with A d,k ( X n ) throughthe canonical isomorphisms, it follows from Proposition 2.1 that we have an iso-morphism χ d,n,k : B d,k ( n ) ∼ = −→ gr k ( A d ( n )) . Thus, we have an isomorphism θ d,n,k := χ d,n,k − . (cid:3) For d ≥
0, we define another functor (cid:102) B d : F op → gVect as follows. For an object n ≥
0, let (cid:102) B d ( n ) := B d ( n ). For a morphism f : m → n in F op , let (cid:102) B d ( f ) := θ d,n ◦ gr( A d ( f )) ◦ θ − d,m : B d ( m ) → B d ( n ) . The family of morphisms θ d := ( θ d,n ) n ≥ : gr ◦ A d ⇒ (cid:102) B d is a natural isomorphism.This is because the PBW maps θ d,m and θ d,n are isomorphisms and because thefollowing diagram commutes:gr( A d ( m )) gr( A d ( f )) (cid:47) (cid:47) ∼ = θ d,m (cid:15) (cid:15) gr( A d ( n )) θ d,n ∼ = (cid:15) (cid:15) B d ( m ) (cid:102) B d ( f ) (cid:47) (cid:47) B d ( n ) . (cid:8) Proposition 3.2.
For d ≥ , we have (cid:102) B d = B d ◦ ab op . Thus, the family of thePBW maps θ d can be rewritten as a natural isomorphism θ d : gr ◦ A d ∼ = ⇒ B d ◦ ab op .In diagram, we have F op A d (cid:47) (cid:47) ab op (cid:15) (cid:15) fVect gr (cid:15) (cid:15) FAb op B d (cid:47) (cid:47) gVect . ∼ = (cid:119)(cid:127) θ d Proof.
We show that (cid:102) B d = B d ◦ ab op . For an element f ∈ F op ( m, n ) = F ( n, m ), let˜ a i,j ∈ N (resp. a i,j ∈ Z ) be the number (resp. the sum of signs) of copies of x ± i that appear in the word f ( x j ) for i = 1 , · · · , m and j = 1 , · · · , n. For example, if f : F → F is defined by(6) f ( x ) = x x x − , f ( x ) = x − x , then the corresponding matrices (˜ a i,j ) and ( a i,j ) are(˜ a i,j ) = (cid:18) (cid:19) , ( a i,j ) = (cid:18) −
11 1 (cid:19) . Note that the matrix A = ( a i,j ) ∈ Mat( m, n ; Z ) corresponds to a morphismab op ( f ) ∈ FAb op ( m, n ). CTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON JACOBI DIAGRAMS 15
For a diagram u (cid:48) = u ( w , · · · , w d − k ) ∈ B d,k ( m ), we prove that (cid:102) B d ( f )( u (cid:48) ) = B d ◦ ab op ( f )( u (cid:48) ) . It suffices to prove the case where w l = v i l for l ∈ [2 d − k ] bymultilinearity. By the definition of the map B d , we have B d ◦ ab op ( f )( u ( v i , · · · , v i d − k )) = u ( v i · A, · · · , v i d − k · A ) . By Lemma 3.1, θ − d,m ( u (cid:48) ) is obtained from u by attaching the l -colored univalentvertex of u to the i l -th component of X m for l ∈ [2 d − k ]. We consider the imageof θ − d,m ( u (cid:48) ) under the map gr( A d ( f )). First, we take a look at an example. For themorphism f : F → F defined by (6), we havegr( A ( f ))( θ − , ( v v )) = gr( A ( f ))( 1 2 )= 1 2 = 1 2 − ∈ gr ( A (2)) . The first term vanishes because two Jacobi diagrams with the same uni-trivalentgraph and with different ways of attaching univalent vertices to a component of X are equivalent in gr ( A (2)) by the STU relations. Thus, we havegr( A ( f ))( θ − , ( v v )) = − . As we observed in the example, the map gr( A d ( f )) sends θ − d,m ( u (cid:48) ) to a linear com-bination of diagrams which are obtained from u by attaching univalent vertices to X n . In particular, the map gr( A d ( f )) sends the l -colored univalent vertex of u tothe signed sum of ˜ a i l ,j copies of the vertex which are attached to the j -th compo-nent of X n for any j = 1 , · · · , n . In the associated graded vector space gr k ( A d ( n )),the image of the l -colored vertex is actually the signed sum of | a i l ,j | copies of thevertex which are attached to the j -th component of X n .Through the PBW map θ d,n again, the Jacobi diagram of (cid:102) B d ( f )( u (cid:48) ) is u . Thecoloring of (cid:102) B d ( f )( u (cid:48) ) that corresponds to the l -colored univalent vertex of u is (cid:80) nj =1 a i l ,j v j = v i l · A , which is equal to that of B d ◦ ab op ( f )( u (cid:48) ). (cid:3) The functors A and B In this section, we compute the functors A and B , and consider the GL( V n )-module structure of B ( n ).Since A , ( n ) = 0, we have A ( n ) = A , ( n ) = gr( A ( n )) θ ,n −−→ ∼ = B ( n ) = B , ( n ) via the PBW map. B ( n ) has a basis { d i,j = v i v j : 1 ≤ i ≤ j ≤ n } and A ( n ) has a basis { c i,j : 1 ≤ i ≤ j ≤ n } corresponding to { d i,j } , where c i,j = i n ( i = j )1 i j n ( i < j ) . Considering the target categories of the functors A and B as the category Vect of vector spaces over k , we have A ∼ = B ◦ ab op by Proposition 3.2. Therefore, it suffices to consider the functor B .We can easily compute the functor B . We extend the notation d i,j by letting d j,i := d i,j for i < j and let D := ( d i,j ) ∈ Mat( n, n ; B ( n )). For a morphism P ∈ FAb op ( m, n ) = Mat( m, n ; Z ), it is easily checked that B ( P )( d i,j ) = ( P D t P ) i,j for 1 ≤ i ≤ j ≤ m .Lastly, we consider the GL( V n )-module structure of B ( n ). Let Sym : Vect → Vect denote the functor that maps a vector space V to Sym ( V ). There is anisomorphism of vector spaces(7) B ( n ) ∼ = −→ Sym ( V n )which maps d i,j to v i · v j for i ≤ j . We associate the functor B to Sym . Definea functor T : FAb op → Vect as follows. For an object n ≥
0, let T ( n ) = V n . Fora morphism P ∈ FAb op ( m, n ) = Mat( m, n ; Z ), let T ( P ) = t P ∈ Mat( n, m ; k ) ∼ =Hom( V m , V n ). We have B ∼ = Sym ◦ T. By the correspondence of the functors B and Sym , we obtain the GL( V n )-module structure of B ( n ). Proposition 4.1.
The linear isomorphism (7) gives a
GL( V n ) -module isomor-phism. Action of gr(IA( n )) on B d ( n )The functor A d gives an Aut( F n ) op -action on A d ( n ), where Aut( F n ) op denotesthe opposite group of Aut( F n ). We have a right Aut( F n )-action on A d ( n ) by letting u · g := A d ( g )( u )for u ∈ A d ( n ) and g ∈ Aut( F n ).We first consider the case n = 1. We haveAut( F ) = { , s } ∼ = GL(1; Z ) ∼ = Z / Z . The action of s on B d,k (1) is multiplication by ( − d − k = ( − k , although it isknown that B d,k (1) = 0 for d ≤ k and it is open whether or not we have B d,k (1) = 0 for all odd k [2].Let IA( n ) denote the IA-automorphism group of F n , which is the kernel of thecanonical homomorphism Aut( F n ) → Aut( H ( F n ; Z )) ∼ = GL( n ; Z ). In this sec-tion, we construct an action of the associated graded Lie algebra gr(IA( n )) of the CTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON JACOBI DIAGRAMS 17 lower central series of IA( n ) on the graded vector space B d ( n ), consisting of grouphomomorphisms β rd,k : gr r (IA( n )) → Hom( B d,k ( n ) , B d,k + r ( n ))for k ≥ r ≥
1, which we define in Section 5.3. In Section 5.5, we extend thisaction by adding the case where r = 0, to obtain an action of an extended gradedLie algebra gr(Aut( F n ) op ) on the graded vector space B d ( n ).5.1. Out( F n ) -action on A d ( n ) . The inner automorphism group
Inn( F n ) of F n isthe normal subgroup of Aut( F n ) consisting of automorphisms σ a for any a ∈ F n ,defined by σ a ( x ) = axa − for x ∈ F n . By the definitions of Inn( F n ) and IA( n ),it follows that Inn( F n ) is a normal subgroup of IA( n ) for any n ≥
1. Here, weconsider the Inn( F n )-action on A d ( n ). Theorem 5.1.
The
Inn( F n ) -action on A d ( n ) is trivial for any d, n ≥ . Therefore,the Aut( F n ) -action on A d ( n ) induces an action on A d ( n ) of the outer automor-phism group Out( F n ) = Aut( F n ) / Inn( F n ) of F n .Thus, the functor A d is an outer functor in the sense of [21] for any d ≥ .Proof. We show that the Inn( F n )-action on A d ( n ) is trivial. Since Inn( F ) =Inn( F ) = 1 and since A ( n ) = k ∅ , we have only to consider for n ≥ , d ≥ u · σ x = u for any u ∈ A d ( n ). Since F n ∼ = Inn( F n ) for n ≥
2, the inner automorphism groupInn( F n ) is generated by σ x , · · · , σ x n . For each i = 2 , · · · , n , define P ,i ∈ Aut( F n )by P ,i ( x ) = x i , P ,i ( x i ) = x , P ,i ( x j ) = x j ( j (cid:54) = 1 , i ) . Then, by (8), we have u · σ x i = u · P ,i σ x P ,i = ( u · P ,i ) · σ x P ,i = u · P ,i P ,i = u. Therefore, we need to prove (8).
For u = 1 2 nu ∈ A d,k ( n ), we have u · σ x = 1 2 nu = 1 2 n · · · u = 1 2 n · · · u = 1 2 n · · · u = u. This completes the proof. (cid:3) n ) -action on A d ( n ) for n = 1 , . For n = 1, we have IA(1) = 1. Therefore,the IA(1)-action on A d (1) is trivial.We use the following fact due to Nielsen [18] and Magnus [16]. See also [17] forthe statement. Theorem 5.2 (Nielsen ( n ≤ n )) . Let n ≥ . The IA-automorphism group IA( n ) is normally generated in Aut( F n ) by an element K , defined by K , ( x ) = x x x − , K , ( x j ) = x j for j (cid:54) = 2 . For n = 2, we have IA(2) = Inn( F ) by the above theorem. Therefore, we havethe following corollary. Corollary 5.3.
The
IA(2) -action on A d (2) is trivial for any d ≥ . Therefore, the Aut( F ) -action on A d (2) induces an action of GL(2; Z ) on A d (2) . Bracket map [ · , · ] : B d,k ( n ) ⊗ Z gr r (IA( n )) → B d,k + r ( n ) . We define(9) [ · , · ] : A d ( n ) × IA( n ) → A d ( n )by [ u, g ] := u · g − u for u ∈ A d ( n ), g ∈ IA( n ), which we call the bracket map . Lemma 5.4.
Let k ≥ . We have [ A d,k ( n ) , IA( n )] ⊂ A d,k +1 ( n ) . Proof.
If we identify the associated graded vector space gr( A d ( n )) with B d ( n ), thenProposition 3.2 implies that the Aut( F n )-action on A d ( n ) induces the GL( n ; Z )-action on B d ( n ). It follows that the restriction of the Aut( F n )-action on gr( A d ( n ))to IA( n ) is trivial. This implies that [ A d,k ( n ) , IA( n )] ⊂ A d,k +1 ( n ). CTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON JACOBI DIAGRAMS 19
Alternatively, we have a direct proof. If we have(10) [ u, K , ] ∈ A d,k +1 ( n ) for all u ∈ A d,k ( n ) , then for any g ∈ Aut( F n ), we have[ u, gK , g − ] = u · gK , g − − u = (( u · g ) · K , − u · g ) · g − = [ u · g, K , ] · g − ∈ A d,k +1 ( n ) . Thus, it suffices to prove (10) by Theorem 5.2.For u = 1 2 nu ∈ A d,k ( n ), we have u · K , = 1 2 nu = 1 2 n · · · u . The summand of box notations such that all dashed lines connected to boxes areattached to the solid line is just u . The other summands are included in A d,k +1 ( n ).Therefore, we have [ u, K , ] = u · K , − u ∈ A d,k +1 ( n ). (cid:3) The following lemma easily follows from the definition of the bracket map.
Lemma 5.5. (1) For g, h ∈ IA( n ) , u ∈ A d,k ( n ) , we have [ u, gh ] = [ u, g ] + [ u, h ] + [[ u, g ] , h ] . (2) For g ∈ IA( n ) , u ∈ A d,k ( n ) , we have [ u, g − ] = − [ u, g ] − [[ u, g ] , g − ] . Proposition 5.6.
The bracket map (9) induces a map (11) [ · , · ] : B d,k ( n ) × IA( n ) → B d,k +1 ( n ) . The map β d,k : IA( n ) → Hom( B d,k ( n ) , B d,k +1 ( n )) defined by β d,k ( g )( u ) = [ u, g ] for g ∈ IA( n ) , u ∈ B d,k ( n ) is a group homomorphism.Proof. By Lemma 5.4, the map (9) induces a map[ · , · ] : gr k ( A d, ∗ ( n )) × IA( n ) → gr k +1 ( A d, ∗ ( n )) . By identifying gr k ( A d, ∗ ( n )) with B d,k ( n ) via the PBW map, we have a map (11).Since we have [ u, gh ] = [ u, g ] + [ u, h ] + [[ u, g ] , h ] and [[ u, g ] , h ] ∈ A d,k +2 ( n ) for g, h ∈ IA( n ) , u ∈ A d,k ( n ) by Lemmas 5.4 and 5.5, it follows that β d,k ( gh )( u ) = [ u, gh ] = [ u, g ] + [ u, h ] = β d,k ( g )( u ) + β d,k ( h )( u )for g, h ∈ IA( n ) , u ∈ B d,k ( n ), so the map β d,k is a group homomorphism. (cid:3) Now we consider the lower central series Γ ∗ (IA( n )) of IA( n ):IA( n ) = Γ (IA( n )) (cid:66) Γ (IA( n )) (cid:66) · · · , where Γ r +1 (IA( n )) = [Γ r (IA( n )) , IA( n )] for r ≥
1. Note that the commutatorbracket [ x, y ] of x and y is defined to be [ x, y ] := xyx − y − for elements x, y of agroup. Let gr(IA( n )) := (cid:76) r ≥ gr r (IA( n )) = (cid:76) r ≥ Γ r (IA( n )) / Γ r +1 (IA( n )) denotethe associated graded Lie algebra with respect to the lower central series of IA( n ).We improve the bracket map (11) and the map β d,k by restricting the maps to thelower central series. Lemma 5.7.
Let r ≥ . We have (12) [ A d,k ( n ) , Γ r (IA( n ))] ⊂ A d,k + r ( n ) . Proof.
We prove (12) by induction on r . The case r = 1 is Lemma 5.4. Supposethat (12) holds for r − ≥
1. By Lemma 5.5, we have[ u, [ g, h ]] = u · ( ghg − h − ) − u = ( u · gh − u · hg ) · g − h − = ([ u, gh ] − [ u, hg ]) g − h − = ([[ u, g ] , h ] − [[ u, h ] , g ]) · g − h − for any g ∈ Γ r − (IA( n )) and h ∈ IA( n ). From the induction hypothesis andLemma 5.4, we have [ u, [ g, h ]] ∈ A d,k + r ( n ). Therefore, by Lemma 5.5, we have[ u, g ] ∈ A d,k + r ( n ) for any g ∈ Γ r (IA( n )) and u ∈ A d,k ( n ). (cid:3) Since A d, d − ( n ) = 0 for any d ≥ n ≥
0, we have the following corollary.
Corollary 5.8.
The
Aut( F n ) -action on A d ( n ) induces an Aut( F n ) / Γ d − (IA( n )) -action on A d ( n ) for d ≥ . We have a canonical isomorphism GL( n ; Z ) ∼ = Aut( F n ) / IA( n ) by the definitionof IA( n ). For any r ≥
1, the abelian group gr r (IA( n )) is a right GL( n ; Z )-moduleby the action induced from the adjoint action of Aut( F n ) on IA( n ). Proposition 5.9.
Let r ≥ . The bracket map (9) induces a GL( n ; Z ) -modulehomomorphism (13) [ · , · ] : B d,k ( n ) ⊗ Z gr r (IA( n )) → B d,k + r ( n ) . Proof.
By Lemmas 5.5 and 5.7, we have a Z -bilinear map[ · , · ] : B d,k ( n ) × gr r (IA( n )) → B d,k + r ( n )and thus, we have a k -linear map (13).Since we have [ u, h ] · g = [ u · g, g − hg ] for g ∈ Aut( F n ) , h ∈ Γ r (IA( n )) , u ∈ A d ( n ),it follows that the following diagram commutes: B d,k ( n ) ⊗ Z gr r (IA( n )) · A (cid:15) (cid:15) [ · , · ] (cid:47) (cid:47) B d,k + r ( n ) · A (cid:15) (cid:15) B d,k ( n ) ⊗ Z gr r (IA( n )) [ · , · ] (cid:47) (cid:47) B d,k + r ( n ) , for any A ∈ GL( n ; Z ). Therefore, (13) is a right GL( n ; Z )-module map. (cid:3) We define a group homomorphism(14) β rd,k : gr r (IA( n )) → Hom( B d,k ( n ) , B d,k + r ( n ))by β rd,k ( g )( u ) = [ u, g ] for g ∈ gr r (IA( n )) and u ∈ B d,k ( n ). CTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON JACOBI DIAGRAMS 21
Actions of extended graded Lie algebras on graded vector spaces.
Webriefly review the definition of extended N-series and extended graded Lie algebrasdefined in [9]. We see that the functor A d gives an action of an extended N-seriesAut ∗ ( F n ) op on the filtered vector space A d, ∗ ( n ), that the action induces an actionof an extended graded Lie algebra gr(Aut( F n ) op ) on the graded vector space B d ( n )and that the induced action is given by the functor B d with morphisms β rd,k .An extended N-series K ∗ = ( K n ) n ≥ of a group K is a descending series K = K ≥ K ≥ K ≥ · · · such that [ K n , K m ] ≤ K n + m for all n, m ≥
0. A morphism f : G ∗ → K ∗ betweenextended N-series is a group homomorphism f : G → K such that we have f ( G n ) ⊂ K n for all n ≥ . For a filtered vector space W ∗ , setAut ( W ∗ ) := Aut fVect ( W ∗ ) , Aut n ( W ∗ ) := { φ ∈ Aut ( W ∗ ) : [ φ, W k ] ⊂ W k + n for all k ≥ } ( n ≥ , where [ φ, w ] := φ ( w ) − w for w ∈ W k . We can easily check that Aut ∗ ( W ∗ ) :=(Aut n ( W ∗ )) n ≥ is an extended N-series. Definition 5.10. (Action of extended N-series on filtered vector spaces) Let K ∗ be an extended N-series and W ∗ be a filtered vector space. An action of K ∗ on W ∗ is a morphism f : K ∗ → Aut ∗ ( W ∗ ) between extended N-series.An extended graded Lie algebra (abbreviated as eg-Lie algebra) L • = ( L n ) n ≥ isa pair of • a graded Lie algebra L + = (cid:76) n ≥ L n , • a group L acting on L + in a degree-preserving way.A morphism f • = ( f n : L n → L (cid:48) n ) n ≥ : L • → L (cid:48)• between eg-Lie algebras consistsof • a group homomorphism f : L → L (cid:48) , • a graded Lie algebra homomorphism f + = ( f n ) n ≥ : L + → L (cid:48) + ,such that we have f n ( x y ) = f ( x ) ( f n ( y )) for n ≥ , x ∈ L and y ∈ L n .We have a functor gr • from the category of extended N-series to the category ofeg-Lie algebras, which maps an extended N-series K ∗ to an eg-Lie algebra gr • ( K ∗ ) =( K /K , (cid:76) n ≥ K n /K n +1 ), where Lie bracket is given by the commutator and theaction of K /K on (cid:76) n ≥ K n /K n +1 is given by the adjoint action.For a graded vector space W = (cid:76) k ≥ W k , setEnd ( W ) := Aut gVect ( W ) , End n ( W ) := { φ ∈ End( W ) : φ ( W k ) ⊂ W k + n for k ≥ } ( n ≥ . We can check that End • ( W ) = (Aut gVect ( W ) , (cid:76) n ≥ End n ( W )) is an eg-Lie alge-bra, where the Lie bracket is defined by[ f, g ] := f ◦ g − g ◦ f for f ∈ End k ( W ) , g ∈ End l ( W )and the action of Aut gVect ( W ) on (cid:76) n ≥ End n ( W ) is defined by the adjoint action g f := g ◦ f ◦ g − for g ∈ Aut gVect ( W ) , f ∈ End k ( W ) . Definition 5.11. (Action of graded Lie algebras on graded vector spaces) Let L + = (cid:76) n ≥ L n be a graded Lie algebra and W = (cid:76) k ≥ W k be a graded vectorspace. An action of L + on W is a morphism f : L + → (cid:76) n ≥ End n ( W ) betweengraded Lie algebras. Definition 5.12. (Action of eg-Lie algebras on graded vector spaces) Let L • bean eg-Lie algebra and W = (cid:76) k ≥ W k be a graded vector space. An action of L • on W is a morphism f : L • → End • ( W ) between eg-Lie algebras. Proposition 5.13.
Let an extended N-series K ∗ act on a filtered vector space W ∗ .Then we have an action of the eg-Lie algebra gr • ( K ∗ ) on the graded vector space gr( W ∗ ) as follows. The group homomorphism ρ : gr ( K ∗ ) → Aut gVect (gr( W ∗ )) is defined by ρ ( gK )([ v ] W k +1 ) = [ g ( v )] W k +1 for gK ∈ gr ( K ∗ ) , and the graded Liealgebra homomorphism ρ + : (cid:77) n ≥ gr n ( K ∗ ) → (cid:77) n ≥ End n (gr( W ∗ )) is defined by ρ + ( gK n +1 )([ v ] W k +1 ) = [[ g, v ]] W k + n +1 for gK n +1 ∈ gr n ( K ∗ ) .Proof. Firstly, we prove that there is a well-defined group homomorphism ρ . Since K ∗ acts on W ∗ , we have a group homomorphism K → Aut(gr k ( W ∗ )) . Moreover, since [ g ( v )] W k +1 = [[ g, v ] + v ] W k +1 = [ v ] W k +1 for g ∈ K and v ∈ W k , itfollows that K → Aut(gr k ( W ∗ )) is trivial. Thus, the group homomorphism ρ : gr ( K ∗ ) = K /K → Aut gVect (gr( W ∗ ))is induced.Secondly, we prove that there is a well-defined Lie algebra homomorphism ρ + .Since K ∗ acts on W ∗ , we can check that ρ + is well defined. Moreover, the map ρ + is a Lie algebra homomorphism because for g ∈ K n , h ∈ K n (cid:48) and v ∈ W k , we have[[ gK n +1 , hK n (cid:48) +1 ] , [ v ] W k +1 ] = [[ g, h ] K n + n (cid:48) +1 , [ v ] W k +1 ]= [[ g, h ] , v ] W k + n + n (cid:48) +1 = [[ g, [ h, v ]] − [ h, [ g, v ]]] W k + n + n (cid:48) +1 =[ gK n +1 , [ hK n (cid:48) +1 , [ v ] W k +1 ]] − [ hK n (cid:48) +1 , [ gK n +1 , [ v ] W k +1 ]] . Lastly, we check that ρ + is compatible with ρ . For g ∈ K , h ∈ K n and v ∈ W k ,we have ρ + ( gK hK n +1 )([ v ] W k +1 ) = [[ ghg − , v ]] W k + n +1 = [ g ([ h, g − ( v )])] W k + n +1 = ρ ( gK ) ( ρ + ( hK n +1 ))([ v ] W k +1 ) . Therefore, this is an action of gr • ( K ∗ ) on gr( W ∗ ). (cid:3) CTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON JACOBI DIAGRAMS 23
Action of gr(IA( n )) on B d ( n ) . Here we observe that the maps β rd,k definedin Section 5.3 give an action of gr(IA( n )) on B d ( n ) in the sense of Definition 5.11.Set Aut r ( F n ) := (cid:40) Aut( F n ) ( r = 0)Γ r (IA( n )) ( r ≥ . The descending series Aut ∗ ( F n ) = (Aut r ( F n )) r ≥ is an extended N-series, so thedescending series Aut ∗ ( F n ) op := (Aut r ( F n ) op ) r ≥ of opposite groups of Aut r ( F n )is also an extended N-series. Let gr(Aut( F n ) op ) denote the image of Aut ∗ ( F n ) op under the functor gr • . We have an action of Aut ∗ ( F n ) op on the filtered vector space A d, ∗ ( n ): A d : Aut( F n ) op → Aut fVect ( A d, ∗ ( n )) . Theorem 5.14.
There is an action of the eg-Lie algebra gr(Aut( F n ) op ) on thegraded vector space B d ( n ) , which can be defined in terms of the functor B d and thegroup homomorphisms β rd,k in (14).Proof. By Proposition 5.13, the action of Aut ∗ ( F n ) op on A d, ∗ ( n ) induces an actionof the eg-Lie algebra gr(Aut( F n ) op ) on the graded vector space gr( A d, ∗ ( n )) ∼ = B d ( n ).This action is a pair of a group homomorphism B d : GL( n ; Z ) op → Aut gVect ( B d ( n ))and a graded Lie algebra homomorphism (cid:77) r ≥ (gr r (Aut ∗ ( F n ) op )) → (cid:77) r ≥ End r ( B d ( n )) , which can be regarded as the group homomorphisms β rd,k by considering the actionof gr r (Aut ∗ ( F n ) op ) on B d ( n ) as a right action of gr r (IA( n )) on B d ( n ). (cid:3) Corollary 5.15.
We have an action of the graded Lie algebra gr(IA( n )) on thegraded vector space B d ( n ) , which consists of the group homomorphisms β rd,k in(14).
6. Aut( F n ) -module structure of A ( n ) and indecomposabledecomposition of the functor A In this section, we study the Aut( F n )-module structure of A ( n ) and give anindecomposable decomposition of the functor A .6.1. Irreducible decomposition of the
GL( V n ) -module B ( n ) . For simplicity,we write V = V n and B d,k = B d,k ( n ).Let N be a nonnegative integer and λ be a partition of N . Let c λ ∈ k S N denotethe Young symmetrizer. Let S λ = k S N · c λ denote the Specht module correspondingto λ , which is a simple S N -module. Let S λ V = V ⊗ N · c λ denote the image of V under the Schur functor S λ , which is a GL( V )-module. Let r ( λ ) be the number ofrows of λ . If r ( λ ) ≤ N , then the GL( V )-module S λ V (cid:54) = 0 is simple. If r ( λ ) > N ,we have S λ V = 0. See [6, 5] for some basic facts about the representation theoryof GL( n ; Z ) and GL( V ).Let B cd,k ⊂ B d,k be the connected part of B d,k , which is spanned by connected V -colored open Jacobi diagrams, and D cd,k ⊂ D d,k the connected part of D d,k , which is spanned by connected special [2 d − k ]-colored open Jacobi diagrams. The subspace D cd,k is an S d − k -submodule of D d,k . We have an isomorphism of GL( V )-modules(15) B cd,k ∼ = V ⊗ d − k ⊗ k S d − k D cd,k , which is the connected version of (4). Proposition 6.1 (Bar-Natan [4]) . We have isomorphisms of S d − k -modules D c , ∼ = S (1 , , , D c , ∼ = S (2) . Proposition 6.2.
We have B = B , ⊕ B , ⊕ B , , where B , ∼ = S (4) V ⊕ S (2 , V,B , = B c , ∼ = S (1 , , V,B , = B c , ∼ = S (2) V as GL( V ) -modules.Proof. The cases where k = 1 , V )-modulesΦ : B , ∼ = −→ Sym ( B c , )defined by Φ( w w w w ) = w w · w w for w , · · · , w ∈ V ,by plethysm, it follows that B , ∼ = Sym ( B c , ) ∼ = S (2) ( S (2) V ) ∼ = S (4) V ⊕ S (2 , V. (cid:3) Let B (cid:48) , (resp. B (cid:48)(cid:48) , ) denote the subspace of B , that is isomorphic to S (4) V (resp. S (2 , V ). By Proposition 6.2, we have an irreducible decomposition of theGL( V )-module B ( n ) B = B (cid:48) , ⊕ B (cid:48)(cid:48) , ⊕ B , ⊕ B , . (16)Here B (cid:48)(cid:48) , vanishes only when n = 0 , B , vanishes only when n = 0 , ,
2, and B (cid:48) , and B , vanish only when n = 0. By (5), the GL( n ; Z )-action on B d ( n ) factorsthrough the GL( V n )-action on B d ( n ):GL( n ; Z ) (cid:44) → GL( n ; k ) t ( · ) − −−−→ GL( V n ) → Aut gVect ( B d ( n )) . Therefore, the irreducible decomposition (16) of B holds as the GL( n ; Z )-modules. Remark . For n = 2, S (2 , V ∼ = det = k is the simple GL(2; Z )-module givenby the square of the determinant, so it is trivial. For n = 3, S (1 , , V ∼ = det is thesimple GL(3; Z )-module given by the determinant.6.2. Direct decomposition of A . Here we give a direct decomposition of thefunctor A .The category A has morphisms µ = : 2 → , η = : 0 → . CTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON JACOBI DIAGRAMS 25
Let c = 1 2 ∈ A (2) and depict it as . The iterated multiplications µ [ q ] : q → q ≥ µ [0] = η, µ [1] = id , µ [ q +1] = µ ( µ [ q ] ⊗ id ) ( q ≥ . For m ≥
0, there is a group homomorphism S m → A ( m, m ) , σ (cid:55)→ P σ , where P σ is the symmetry in A corresponding to σ . Set sym m := (cid:88) σ ∈ S m P σ , alt m := (cid:88) σ ∈ S m sgn( σ ) P σ ∈ A ( m, m ) . By Habiro–Massuyeau [10, Lemma 5.16], every element of A ( n ) is a linearcombination of morphisms of the form( µ [ q ] ⊗ · · · ⊗ µ [ q n ] ) ◦ P σ ◦ c ⊗ for σ ∈ S and q , · · · , q n ≥ q + · · · + q n = 4. The following lemmaeasily follows. Lemma 6.4.
For n ≥ , we have A ( n ) = Span k { A ( f )( c ⊗ c ) : f ∈ F op (4 , n ) } . Set P (cid:48) = sym , P (cid:48)(cid:48) = alt alt ∈ A (4) . Let A (cid:48) (resp. A (cid:48)(cid:48) ) : F op → fVect be the subfunctor of the functor A such that A (cid:48) ( n ) := Span k { A ( f )( P (cid:48) ) : f ∈ F op (4 , n ) } ⊂ A ( n )(resp. A (cid:48)(cid:48) ( n ) := Span k { A ( f )( P (cid:48)(cid:48) ) : f ∈ F op (4 , n ) } ⊂ A ( n )) . Proposition 6.5.
We have a direct decomposition A = A (cid:48) ⊕ A (cid:48)(cid:48) in the functor category fVect F op .Proof. We prove that A ( n ) = A (cid:48) ( n ) + A (cid:48)(cid:48) ( n ) for n ≥
0. Since we have sym + 4 alt alt + 4 alt alt = (8 c ⊗ c + 8 + 8 ) + (8 c ⊗ c − c ⊗ c − c ⊗ c, it follows that c ⊗ c ∈ A (cid:48) ( n ) + A (cid:48)(cid:48) ( n ). Thus, we have A ( n ) = A (cid:48) ( n ) + A (cid:48)(cid:48) ( n ) byLemma 6.4.In order to prove that A (cid:48) ( n ) ∩ A (cid:48)(cid:48) ( n ) = 0, it suffices to show that θ ,n (gr( A (cid:48) ( n ))) ⊂ B (cid:48) , ( n ) , θ ,n (gr( A (cid:48)(cid:48) ( n ))) ⊂ B (cid:48)(cid:48) , ( n ) ⊕ B , ( n ) ⊕ B , ( n ) . Let P (cid:48) ijkl = sym i j k l be a linear sum of elements of A ( n ) such that eachendpoint of two chords are attached to the i, j, k, l -th component of X n , respectively,where 1 ≤ i ≤ j ≤ k ≤ l ≤ n . Note that P (cid:48) ijkl is defined independently of how toattach endpoints to the same component of X n because of the symmetrizer. Sincean element of A (cid:48) ( n ) is a linear sum of P (cid:48) ijkl and since we have θ ,n ( P (cid:48) ijkl ) = sym v i v j v k v l ∈ B (cid:48) , ( n ) , it follows that θ ,n (gr( A (cid:48) ( n ))) ⊂ B (cid:48) , ( n ).Let P (cid:48)(cid:48) ijkl = alt alt i j k l be a linear sum of elements of A ( n ) such that each end-point of two chords are attached to the i, j, k, l -th component, respectively, where i, j, k, l ∈ { , · · · , n } . Note that P (cid:48)(cid:48) ijkl has ambiguity of how to attach endpoints tothe same component of X n , but the difference is an element of A , ( n ). Since anelement of A (cid:48)(cid:48) ( n ) is a linear sum of P (cid:48)(cid:48) ijkl and since we have θ ,n ( P (cid:48)(cid:48) ijkl ) − alt alt v i v j v k v l ∈ B , ( n ) ⊕ B , ( n )by Lemma 3.1, it follows that θ ,n (gr( A (cid:48)(cid:48) ( n ))) ⊂ B (cid:48)(cid:48) , ( n ) ⊕ B , ( n ) ⊕ B , ( n ) . (cid:3) Proposition 6.6.
We have θ ,n (gr( A (cid:48) ( n ))) = B (cid:48) , ( n ) ,θ ,n (gr( A (cid:48)(cid:48) ( n ))) = B (cid:48)(cid:48) , ( n ) ⊕ B , ( n ) ⊕ B , ( n ) . (17) Proof.
This follows from Proposition 6.5 and Lemma 3.1. (cid:3)
Action of gr(IA( n )) on B ( n ) . In order to study the Aut( F n )-module struc-ture of A (cid:48)(cid:48) ( n ), we consider whether the restrictions of the GL( n ; Z )-module homo-morphism (13)(18) [ · , · ] : B (cid:48)(cid:48) , ( n ) ⊗ Z gr (IA( n )) → B , ( n ) , (19) [ · , · ] : B , ( n ) ⊗ Z gr (IA( n )) → B , ( n )vanish or not.For n = 1 ,
2, the maps (18) and (19) vanish, because the IA( n )-actions on A ( n )are trivial by Corollary 5.3.The bracket maps (18) and (19) induce GL( n ; Z )-module homomorphisms ρ : B (cid:48)(cid:48) , ( n ) → Hom(gr (IA( n )) , B , ( n )) ,ρ : B , ( n ) → Hom(gr (IA( n )) , B , ( n )) , respectively.For distinct elements i, j, k ∈ [ n ], define K i,j,k ∈ IA( n ) by K i,j,k ( x i ) = x i [ x j , x k ] , K i,j,k ( x l ) = x l for l (cid:54) = i. CTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON JACOBI DIAGRAMS 27
Lemma 6.7.
For n ≥ , the GL( n ; Z ) -module homomorphisms ρ and ρ areinjective.Proof. Let u = v v v v − v v v v ∈ B (cid:48)(cid:48) , ( n ) . We have ρ ( u )( K , , ) = 6 v v v (cid:54) = 0 ∈ B , ( n ) . Thus, we have ρ (cid:54) = 0. Since B (cid:48)(cid:48) , ( n ) is simple, it follows that ρ is injective.We have ρ ( v v v )( K , , ) = v v (cid:54) = 0 ∈ B , ( n ) . Since B , ( n ) is simple, it follows that ρ is injective in a similar way. (cid:3) Remark . The restriction of the GL( n ; Z )-module homomorphism (13)[ · , · ] : B (cid:48)(cid:48) , ( n ) ⊗ Z gr (IA( n )) → B , ( n )also induces a GL( n ; Z )-module homomorphism ρ : B (cid:48)(cid:48) , ( n ) → Hom(gr (IA( n )) , B , ( n )) . We can also check that ρ is injective. This is because we have[ u, [ K , , , K , , ]] = [[ u, K , , ] , K , , ] − [[ u, K , , ] , K , , ]= [6 v v v , K , , ] = 6 v v (cid:54) = 0 ∈ B , ( n )since [ u, K , , ] = 0.6.4. Aut( F n ) -module structure of A ( n ) . Here, we consider the Aut( F n )-modulestructure of A ( n ).By Proposition 6.5, we have a decomposition of Aut( F n )-modules A ( n ) = A (cid:48) ( n ) ⊕ A (cid:48)(cid:48) ( n )and a filtration of Aut( F n )-modules A (cid:48)(cid:48) ( n ) ⊃ A , ( n ) ⊃ A , ( n ) ⊃ . Moreover, by (16) and (17), we have GL( n ; Z )-module isomorphisms θ ,n (gr( A (cid:48) ( n ))) = B (cid:48) , ( n ) ∼ = S (4) V n ,θ ,n (gr( A (cid:48)(cid:48) ( n ))) = B (cid:48)(cid:48) , ( n ) ⊕ B , ( n ) ⊕ B , ( n ) ∼ = S (2 , V n ⊕ S (1 , , V n ⊕ S (2) V n ,θ ,n (gr( A , ( n ))) = B , ( n ) ⊕ B , ( n ) ∼ = S (1 , , V n ⊕ S (2) V n ,θ ,n (gr( A , ( n ))) = B , ( n ) ∼ = S (2) V n . Thus, it follows that A (cid:48) ( n ) and A , ( n ) are simple Aut( F n )-modules for any n ≥ Theorem 6.9.
The
Aut( F n ) -module A (cid:48) ( n ) is simple for any n ≥ .For n = 1 , Aut( F ) ∼ = Z / Z acts on A (cid:48)(cid:48) (1) ∼ = k trivially.For n = 2 , A (cid:48)(cid:48) (2) has an irreducible decomposition A (cid:48)(cid:48) (2) = W ⊕ A , (2) , where Aut( F ) acts on W ∼ = k trivially.For n ≥ , A (cid:48)(cid:48) ( n ) admits a unique composition series of length A (cid:48)(cid:48) ( n ) (cid:41) A , ( n ) (cid:41) A , ( n ) (cid:41) that is, A (cid:48)(cid:48) ( n ) has no nonzero proper Aut( F n ) -submodules other than A , ( n ) and A , ( n ) . Therefore, A (cid:48)(cid:48) ( n ) and A , ( n ) are indecomposable. To prove this theorem, we use the following fact due to Nielsen [19]. See also[17] for the statement. Define U , , P , , σ ∈ Aut( F ) by U , ( x ) = x x , U , ( x ) = x ,P , ( x ) = x , P , ( x ) = x ,σ ( x ) = x − , σ ( x ) = x . Theorem 6.10 (Nielsen [19]) . The automorphism group
Aut( F ) is generated by U , , P , and σ .Proof of Theorem 6.9. For n = 1, A (cid:48)(cid:48) (1) = A , (1) has a basis { } . We cancheck that the action of Aut( F ) ∼ = Z / Z on A (cid:48)(cid:48) (1) is trivial.For n = 2, let u = 2 1 2 − − ∈ A (cid:48)(cid:48) (2) \ A , (2) ,u , = 12 1 2 , u , = 1 2 , u , = 12 21 ∈ A , (2) . It is easily checked that { u, u , , u , , u , } is a basis for A (cid:48)(cid:48) (2) and that the repre-sentation matrices of U , , P , and σ for this basis are U , = , P , = , σ = − . Let w = (1 , − , , −
1) = u − u , − u , ∈ A (cid:48)(cid:48) (2) \ A , (2) and W = k w . We have w · U , = w · P , = w · σ = w. Thus, by Theorem 6.10, it follows that the Aut( F )-action on W is trivial. There-fore, we have an irreducible decomposition A (cid:48)(cid:48) (2) = W ⊕ A , (2) . For n ≥
3, since A (cid:48)(cid:48) ( n ) /A , ( n ) ∼ = B (cid:48)(cid:48) , ( n ) ∼ = S (2 , V n , A , ( n ) /A , ( n ) ∼ = B , ( n ) ∼ = S (1 , , V n are simple Aut( F n )-modules, we have a composition series of length 3 A (cid:48)(cid:48) ( n ) (cid:41) A , ( n ) (cid:41) A , ( n ) (cid:41) . CTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON JACOBI DIAGRAMS 29
We next prove that A , ( n ) does not have any nonzero proper submodules otherthan A , ( n ). (Then, it follows that A , ( n ) is indecomposable.) Let A be a nonzerosubmodule of A , ( n ) other than A , ( n ). Since A , ( n ) is simple, there is an ele-ment a ∈ A \ A , ( n ). We have θ ,n (gr( A , ( n ))) = B , ( n ) ⊕ B , ( n ), so we can write a as a = u + v , for some elements u (cid:54) = 0 ∈ θ − ,n ( B , ( n )), v ∈ θ − ,n ( B , ( n )) = A , ( n ).By Lemma 6.7, there is g ∈ IA( n ) such that [ u, g ] (cid:54) = 0 ∈ A , ( n ). Therefore, wehave [ a, g ] = [ u + v, g ] = [ u, g ] + [ v, g ] = [ u, g ] (cid:54) = 0 ∈ A , ( n ) . Since A , ( n ) is simple, we have A , ( n ) (cid:40) A . Since A , ( n ) has a compositionseries of length 2, by the Jordan–H¨older theorem, we have A = A , ( n ).We now prove that A (cid:48)(cid:48) ( n ) does not have any nonzero proper submodules otherthan A , ( n ) and A , ( n ). (Then, it follows that A (cid:48)(cid:48) ( n ) is indecomposable.) Let A be a nonzero submodule of A (cid:48)(cid:48) ( n ) other than A , ( n ) , A , ( n ). Since A , ( n ) is theonly nonzero proper submodule of A , ( n ), we have A (cid:42) A , ( n ). Thus, there isan element a ∈ A \ A , ( n ). Since we have θ ,n (gr( A (cid:48)(cid:48) ( n ))) = B (cid:48)(cid:48) , ( n ) ⊕ B , ( n ) ⊕ B , ( n ), we can write a as a = u + v for some elements u (cid:54) = 0 ∈ θ − ,n ( B (cid:48)(cid:48) , ( n )), v ∈ θ − ,n ( B , ( n ) ⊕ B , ( n )) = A , ( n ). By Lemma 6.7, there is g ∈ IA( n ) such that[ u, g ] ∈ A , ( n ) \ A , ( n ). Therefore, we have[ a, g ] = [ u + v, g ] = [ u, g ] + [ v, g ] ∈ A , ( n ) \ A , ( n ) , because [ v, g ] ∈ A , ( n ). Since A , ( n ) is the only nonzero proper submodule of A , ( n ), we have A ∩ A , ( n ) = A , ( n ) and therefore, A , ( n ) (cid:40) A . Since A (cid:48)(cid:48) ( n )has a composition series of length 3, by the Jordan–H¨older theorem, we have A = A (cid:48)(cid:48) ( n ). (cid:3) Corollary 6.11.
There are exact sequences of
Aut( F n ) -modules → S (1 , , V n → A (cid:48)(cid:48) ( n ) /A , ( n ) → S (2 , V n → , → S (2) V n → A , ( n ) → S (1 , , V n → , which do not split for n ≥ . Thus, we have Ext k Aut( F n ) ( S (2 , V n , S (1 , , V n ) (cid:54) = 0 , Ext k Aut( F n ) ( S (1 , , V n , S (2) V n ) (cid:54) = 0 for n ≥ .Remark . Corollary 6.11 also holds as Out( F n )-modules. Remark . By Theorem 5.1, the Aut( F n )-module A , ( n ) can be considered asan Out( F n )-module. There is no Out( F )-modules of dimension less than 7 whichdo not factor through the canonical surjection Out( F ) (cid:16) GL(3; Z ) [14]. Turchinand Willwacher [22] constructed the first 7-dimensional Out( F )-module U I withsuch property. We can check that the Out( F )-module A , (3) is isomorphic to U I . We have another 7-dimensional Out( F )-module A (cid:48)(cid:48) (3) /A , (3) which does notfactor through GL(3; Z ). At the level of the associated graded GL(3; Z )-module,gr( A (cid:48)(cid:48) (3) /A , (3)) ∼ = S (2 , V ⊕ S (1 , , V ∼ = ( S (2) V ) ∗ ⊕ ( S (1 , , V ) ∗ ∼ = gr( A , (3)) ∗ . We conjecture that the Out( F )-module A (cid:48)(cid:48) (3) /A , (3) is isomorphic to the dual of A , (3) and that A (cid:48)(cid:48) (3) is self-dual. Indecomposable decomposition of A . Lastly, we show that the subfunc-tors A (cid:48) and A (cid:48)(cid:48) of A , which we observed in Section 6.2, are indecomposable. Theorem 6.14.
The direct decomposition A = A (cid:48) ⊕ A (cid:48)(cid:48) in Proposition 6.5 isindecomposable in the functor category fVect F op .Proof. Since A (cid:48) ( n ) are simple Aut( F n )-modules for any n ≥ A (cid:48) is indecomposable in fVect F op .Suppose that we have a direct decomposition A (cid:48)(cid:48) = G ⊕ G (cid:48) ∈ fVect F op , where G (cid:48) is possibly 0. By Theorem 6.9, the Aut( F n )-modules A (cid:48)(cid:48) ( n ) are indecom-posable for n ≥
3, we have(20) G ( n ) = A (cid:48)(cid:48) ( n ) , G (cid:48) ( n ) = 0 ( n ≥ . Since G and G (cid:48) are subfunctors of A , (20) holds for any n ≥
0. Therefore, we have G = A (cid:48)(cid:48) , G (cid:48) = 0 . This implies that the functor A (cid:48)(cid:48) is indecomposable in fVect F op . (cid:3) Perspectives
In a paper in preparation [12], we plan to study the Aut( F n )-module structureof A d ( n ) and the functor A d for d ≥ V n )-module B d ( n ) and the action of gr(IA( n )) on B d ( n ). For small d ≥
3, theGL( V n )-module structure of B d ( n ) is obtained similarly, but it is rather difficult tocompute the gr(IA( n ))-action on B d ( n ) directly. In order to simplify computationof the gr(IA( n ))-action on B d ( n ), we will reconstruct the action in a different way.We will study the Johnson filtration E ∗ ( n ) of the endomorphism monoid End( F n ),which is an enlargement of the Johnson filtration A ∗ ( n ) of Aut( F n ) and the lowercentral series Γ ∗ (IA( n )) of IA( n ). Here, the r -th term E r ( n ) of the Johnson filtrationis defined to be E r ( n ) := ker(End( F n ) → End( F n / Γ r +1 ( F n ))) , where Γ r +1 ( F n ) denotes the r + 1-st term of the lower central series of F n . We cancheck that E ∗ ( n ) acts on A d ( n ) and therefore the extended N-series A ∗ ( n ) acts onthe filtered vector space A d ( n ).We obtain a graded Lie algebra gr( E ∗ ( n )) and a group isomorphismgr r ( E ∗ ( n )) ∼ = Hom( H, L r +1 ( n )) , where H = H ( U n ; Z ) ∼ = Z n is a free abelian group and L r +1 ( n ) is the degree r + 1part of the free Lie algebra generated by H . In particular, by Andreadakis [1]( n = 3) and by Kawazumi [13] (for any n ), we havegr (IA( n )) ∼ = Hom( H, L ( n )) ∼ = gr ( E ∗ ( n )) . We will construct two equivalent actions of gr r ( E ∗ ( n )) ∼ = Hom( H, L r +1 ( n )) on B d ( n ), one of which is a generalization of the gr(IA( n ))-action and the other ofwhich we can compute easily. This makes it easier to compute the gr (IA( n ))-action on B d ( n ). CTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON JACOBI DIAGRAMS 31
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Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
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